Setwise intersecting families of permutations

A family of permutations A ⊂ Sn is said to be t-set-intersecting if for any two permutations σ, π ∈ A, there exists a t-set x whose image is the same under both permutations, i.e. σ(x) = π(x). We prove that if n is sufficiently large depending on t, the largest t-set-intersecting families of permutations in Sn are cosets of stabilizers of t-sets. The t = 2 case of this was conjectured by János Körner. It can be seen as a variant of the Deza-Frankl conjecture, proved in [6]. Our proof uses similar techniques to those of [6], namely, eigenvalue methods, together with the representation theory of the symmetric group, but the combinatorial part of the proof is harder.